Algebra Problems of the Day (Algebra 1 Regents, June 2023 Part I)

The following questions appeared on the une 2023 Algebra 1 Regents Exam

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Algebra 1 Regents, June 2023

Part I: Each correct answer will receive 2 credits.


9. A company ships an average of 30,000 items each week. The approximate number of items shipped each minute is calculated using the conversion

Answer: 2) [See Image]

To get items per minute, you need to have items in the numerator and minutes in the denominator. Every other unit needs to cancel out (reduce to 1) by appearing once in each of the numerator and the denominator. You can cancel units like factors.

Choice (1) has minutes in the numerator, and two weeks in the denominator, which would give you "weeks²", which is meaningless. Eliminate Choice (1).

Choice (2) has items in the numerator and minutes in the denominator. It also has weeks, days, and hours in both the numerator and the denominator, so they they cancel out. Choice (2) is the correct answer.

Choices (3) and (4) have items in the denominator. Eliminate Choices (3) and (4).

10. A function is graphed below.

A possible equation for this function is

1) f(x) = (x + 2)(x - 3)
2) f(x) = (x - 2)(x + 3)
3) f(x) = (x - 2)²(x + 3)
4) f(x) = (x - 2)(x + 3)(x - 12)

Answer:
3) f(x) = (x - 2)²(x + 3)

The zeroes of the function are at x = -3 and x = 2, which makes (x + 3) and (x - 2) factors. This is a cubic function, so you no that there is a third factor. (In reality, those fact checking me will tell you that it's obviously a function with an odd number exponent greater than 1, not an even number exponent. We aren't dealing with those graphs in Algebra 1, so don't worry about that right now.)

Basically, it's not a parabola, so it's not (1) or (2).

Choice (4) uses the y-intercept for no particular reason except to confuse you. There is no zero at x = 12. The graph indicates that it continues upward.

The fact that there is a minimum point is (2,0) is the reason that there are two factors of (x - 2), which is (x - 2)². If you multiply (-2)(-2)(3), you will get +12, which is the expected y-intercept on the graph.

Choice (3) is the correct answer.

As with similar problems, you could put all four choices into your graphing calculator and see which one looks like the given graph.

11. If g(x) = -x² - x + 5, then g(-4) is equal to?

1) -15
2) -7
3) 17
4) 25

Answer: 4) 25

Put the function in your calculator and look at the table of values for x = -4. Or just enter (-4)² - (-4) + 5. Make sure that you use parentheses.

(-4)² - (-4) + 5 = 16 + 4 + 5 = 25, which is Choice (4).

12. A movie theater’s popcorn box is a rectangular prism with a base that measures 6 inches by 4 inches and has a height of 8 inches. To create a larger box, both the length and the width will be increased by x inches. The height will remain the same. Which function represents the volume, V(x), of the larger box?

1) V(x) = (6 + x)(4 + x)(8 + x)
2) V(x) = (6 + x)(4 + x)(8)
3) V(x) = (6 + x) + (4 + x) + (8 + x)
4) V(x) = (6 + x) + (4 + x) + (8)

Answer: 2) V(x) = (6 + x)(4 + x)(8)

Volume equals length times width times height, so eliminate Choice (3) and (4).

The height of the box doesn't change, so there is no reason to add x to the 8 inches of the original box. Eliminate Choice (1).

Choice (2) show the length and the width increased by the same x inches while the height remains its orignal 8 inches. Choice (2) is the correct answer.

13. The expression 300(4)^{x + 3} is equivalent to

1) 300(4)^x(4)³
2) 300(4^x)³
3) 300(4^x) + 300(4³)
4) 300^x(4³)

Answer: 1) 300(4)^x(4)³

If you mutliply two terms with the same base, you add the exponents. So multiplying (4)^x(4)³ will give you (4)^{x + 3}. Choice (1) is the correct answer.

If you don't believe that, you can try it on your calculator using any number in place of x, or enter both expressions as graphs and look at the matching tables of values.

Choice (2) can be rewritten as 300(4)^3x, which isn't what we are looking for.

Choice (3) can be rewritten as 300(4^x + 4³), which isn't what we want.

Choice (4) is just nutty. It's there for people who don't evenread the questions before they choose an answer. Why would 300 suddenly gain an exponent of x?

14. Ashley only has 7 quarters and some dimes in her purse. She needs at least $3.00 to pay for lunch. Which inequality could be used to determine the number of dimes, d, she needs in her purse to be able to pay for lunch?

1) 1.75 + d > 3.00
2) 1.75 + 0.10d > 3.00
3) 1.75 + d < 3.00
4) 1.75 + 0.10d < 3.00

Answer: 2) 1.75 + 0.10d > 3.00

"At least" means that you want to have an amount that is greater than or equal to $3.00. Eliminate Choice (3) and (4).

Ashley doesn't want to add the number of dimes to the $1.75. She needs to add the value of those dimes, which is $0.10 times the number of dimes, d, or 0.10d.

Choice (2) is the correct answer.

15. The formula for the area of a trapezoid is A = 1/2(b₁ + b₂)h.
The height, h, of the trapezoid may be expressed as

Answer: 4) [See Image]

You can solve a literal equation in the same manner that you solve an equation with one variable. The only difference is that you most likely won't combine any "like terms".

To solve for h, isolate the variable by using inverse operations.

A = 1/2(b₁ + b₂)h

2A = (b₁ + b₂)h

2A / (b₁ + b₂) = h

This is Choice (4).

16. The function f(x) = |x| is multiplied by k to create the new function g(x) = k|x|. Which statement is true about the graphs of f(x) and g(x) if k = 1/2?

1) g(x) is a reflection of f(x) over the y-axis
2) g(x) is a reflection of f(x) over the x-axis.
3) g(x) is wider than f(x).
4) g(x) is narrower than f(x).

Answer: 3) g(x) is wider than f(x).

As with a parabola, if the coefficient is between 0 and 1, the function will be wider than f(x) = |x|. This is Choice (3).

If k = -1, then g(x) would be a reflection of f(x) over the x-axis.

A reflection of f(x) = |x| over the y-axis would be equal to f(x) itself.

If k > 1 (or k < -1) then g(x) would be narrower than f(x).

More to come. Comments and questions welcome.

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